Iskanje po katalogu digitalne knjižnice

Opis: In this doctoral dissertation, we address the counting of Hamiltonian cycles in 2-tiled graphs. These graphs are a generalization of the construction of large 2-crossing-critical graphs. We also address the study of the learning process involved in solving unsolved mathematical problems, integrating psychological theories of optimal experience (flow) and deliberate practice into a mathematical framework called the dense challenge domain. The introduction presents fundamental graph theory concepts and an overview of ordinal numbers essential for understanding the core of the dissertation. In the second chapter, known results from relevant related fields are introduced, along with the contributions of the doctoral dissertation. In the third chapter, we address the problem of counting Hamiltonian cycles in 2-tiled graphs. First, we introduce basic concepts such as tile, 2-tile, tiled graphs, and 2-tiled graphs. This is followed by results leading to the characterization of Hamiltonian cycle types in 2-tiled graphs and then the introduction of algorithms that count each type of Hamiltonian cycles. We also demonstrate that if the family of 2-tiles used to construct 2-tiled graphs is finite, the algorithms are efficient. Further, we place large 2-crossing-critical graphs in the context of 2-tiled graphs and adapt the previously introduced algorithms to efficiently count all types of Hamiltonian cycles. To describe 2-crossing-critical graphs, we introduce an alphabet and show that 2-traversing and flanking Hamiltonian cycles can be counted by simply counting the occurrences of certain letters from the introduced alphabet. At the end of the chapter, we extend the counting of traversing Hamiltonian cycles to tiled graphs. We attempt to formally capture the experience of this research work in the fourth chapter. We propose a formal mathematical framework for solving a common challenge in mathematical education: how to effectively use limited time to motivate students for research work. We formalize a theoretical mathematical structure called the dense challenge domain and introduce a structured decision process based on Csikszentmihalyi’s theory of flow, Duckworth’s concept of grit, Snowden’s Cynefin framework from decision theory, and Bokal-Steinbacher’s time usage optimization model. Unlike traditional educational research, which focuses on primary and secondary education, our approach emphasizes fostering mathematical thinking at the research level through optimal psychological experience. We formalize an algorithm for evolving the dense challenge domain, ensuring a balance between perceived skill and levels of challenge. We further prove that large 2-crossing-critical graphs satisfy the conditions of the dense challenge domain, providing a solid mathematical foundation for this methodology. Through three pilot studies, we demonstrate that this approach not only maintains student engagement but also leads to publishable research outcomes.
Ključne besede: Hamiltonian cycle, 2-crossing-critical graph, 2-tiled graph, research realm, dense challenge domain, psychologically optimal experience, Cynefin framework
Objavljeno v DKUM: 24.10.2025; Ogledov: 0; Prenosov: 4
Celotno besedilo (1,76 MB)

Opis: The results of the presented research are directly applicable to enhancing the maintenance strategies of transmission system operator (TSO) assets, particularly high-voltage switchyard equipment, and are extendable to other TSO systems. Furthermore, the proposed methodology lays the groundwork for the implementation of predictive maintenance. This paper introduces a novel methodology for assessing the technical condition of high-voltage switchyard devices—specifically, circuit breakers, disconnectors, and instrument transformers—within a high-voltage switching bay. The proposed model integrates multiplicative criteria, which reflect maintenance actions, and additive criteria, which capture the operational and maintenance history of individual devices. Supported by a newly developed data model, the methodology enables an automated assessment process that generates a c-curve representing the condition trajectory of each device or device type. Leveraging real-time data from the maintenance information system, this automated approach allows for the timely evaluation of a device’s technical state prior to scheduled maintenance. The resulting c-curve analysis supports strategic maintenance planning and prioritization. The proposed solutions have been implemented experimentally by TSO ELES.
Ključne besede: condition-based maintenance, health index, circuit breaker, disconnector, instrument transformer, substation, failure mode analysis
Objavljeno v DKUM: 23.06.2025; Ogledov: 0; Prenosov: 11
Celotno besedilo (1,52 MB)
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Opis: The masters thesis, which follows the paper Graph drawing applications in combinatorial theory of maturity models, in preparation, coauthored by the author of the thesis, introduces the tiled graphs as models of learning and maturing processes. In the thesis, we show how tiled graphs can combine graphs of learning spaces or antimatroids (partial cubes) and maturity models (total orders) to yield models of learning processes. We visualise processes with optimal drawings. In the thesis, we show NP-hardness of visualisation problems resulting from most detailed models. Further, we introduce a simpler model, which ignores the details of learning and for which the visualisation problem can be solved in a polynomial time. For the rest of the thesis, we consider this model. We describe an algorithm, which finds a drawing of an ordinal panel data graph with a minimal number of edge crossings. For this problem we further define an extremal crossing number for a chosen family of ordinal panel data. Further, we explore a certain type of random instances of ordinal panel data and the expected value of a crossing number for this type of random instances. After that, we define a problem of finding the most suitable ordering on categories in panel data, in other words finding the best maturity model to fit the data. We prove the NP-hardness of the problem and formulate an integer linear program. Master thesis consists of nine chapters. The first chapter contains known results and definitions from set and graph theory and a section of computational complexity theory (NP-hardness), which will be used throughout the thesis. The following chapters present the new theory introduced in the aforementioned paper in preparation and the needed additional results and definitions. In the last chapter we present the thesis and some selected parts of the thesis with the help of learning space theory. The chapter serves as both the overview of the thesis and the use case for the theory of learning spaces, presented in the thesis.
Ključne besede: Maturity models, learning spaces, crossing number, crossing minimisation, tile crossing number.
Objavljeno v DKUM: 14.03.2024; Ogledov: 463; Prenosov: 39
Celotno besedilo (1,46 MB)

Opis: Pursuit-evasion games study the number of cops needed to capture therobber in a game played on a graph, in which the cops and the robber movealternatively to neighbouring vertices, and the robber is captured if a copsteps on the vertex the robber is in. A common tool in analyzing this copnumber of a graph is a cop moving along a shortest path in a graph, thuspreventing the robber to step onto this path. We generalize this approach byintroducing a shadow of the robber, the maximal set of vertices from whichthe cop parries the protected subgraph. In this context, the robber becomesan intruder and the cop becomes the guard. We show that the shadow canbe computed in polynomial time, implying polynomial time algorithms forcomputing both a successful guard as well as a successful intruder, whicheverexists. Furthermore, we show that shadow function generalizes the conceptof graph retractions. In some cases, this implies a polynomially computablecertification of the negative answer to the NP-complete problem of existenceof a retraction to a given subgraph.
Ključne besede: pursuit-evasion game, graph searching, guarding, shadow function, graph retraction
Objavljeno v DKUM: 17.08.2023; Ogledov: 423; Prenosov: 56
Celotno besedilo (377,93 KB)
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Opis: Priročnik primerov pedagoške prakse poučevanja in učenja na univerzitetnem nivoju z osredotočenostjo na naravoslovno-matematična področja je namenjen vsem visokošolskim učiteljem in sodelavcem, ki želijo v svoj pedagoški proces vpeljati preverjene pedagoške prakse z namenom izboljšati študijski uspeh in opolnomočiti študente z ustreznimi kompetencami za njihovo nadaljnjo karierno pot. V priročniku predstavimo proces usvajanja znanja od odkrivanja do uporabe skozi lestvico stopenj zrelosti tehnologij. Pri tem želimo poudariti omejenost virov kot so znanje, inteligenca, čustvena inteligenca, empatija, čas, pozornost, zaupanje in pomen njihovega optimalnega izkoriščanja. Skozi konkretne primere učinkovitih pedagoških praks želimo vzpodbuditi visokošolske učitelje in sodelavce, da v pedagoški proces vpeljejo sodobne metode učenja in jih kombinirajo s tradicionalnimi.
Ključne besede: univerzalni model procesa, tehnološka zrelost, učinkovite pedagoške prakse, naravoslovje, matematika
Objavljeno v DKUM: 16.05.2022; Ogledov: 719; Prenosov: 112
Celotno besedilo (3,07 MB)
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